Fundamental progroupoid and bundles with a structural category

Ardanza-Trevijano, Sergio; Paricio, Luis Javier Hernández

doi:10.1016/s0166-8641(97)00247-2

Cited by 5 publications

(12 citation statements)

References 5 publications

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“…commutes when ev is evaluation ev(y, α) = α (1). Thus p F is continuous by the universal property of quotient spaces.…”

Section: Existence Of Semicoveringsmentioning

confidence: 93%

“…The top map P ζ−1 is, on each summand, the product of open embeddings (by induction hypothesis): (1). Consequently, the lift of δ * starting at y 1 ends at y 2 .…”

Section: Open Covering Morphismsmentioning

confidence: 98%

“…there are open neighborhoods V 0 , V 1 of α(0), α (1) in X respectively such that for every a ∈ V 0 , b ∈ V 1 there is a path β ∈ U with β(0) = a and β(1) = b.…”

Section: A Path α : I → X Is Well-ended If For Every Open Neighborhoomentioning

confidence: 99%

“…6.7,6.8] to determine when certain maps have unique path lifting. We first note that if p F (y) = x and α ∈ Ω(X, x), the coset (1)) such that for each u ∈ U t (resp. v ∈ V t ), there is a well-targeted path δ ∈ A t with δ(1) = u (resp.…”

Section: Proof If Ymentioning

confidence: 99%

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Semicoverings: a generalization of covering space theory

Brazas¹

2012

Homology, Homotopy and Applications

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“…commutes when ev is evaluation ev(y, α) = α (1). Thus p F is continuous by the universal property of quotient spaces.…”

Section: Existence Of Semicoveringsmentioning

confidence: 93%

“…The top map P ζ−1 is, on each summand, the product of open embeddings (by induction hypothesis): (1). Consequently, the lift of δ * starting at y 1 ends at y 2 .…”

Section: Open Covering Morphismsmentioning

confidence: 98%

“…there are open neighborhoods V 0 , V 1 of α(0), α (1) in X respectively such that for every a ∈ V 0 , b ∈ V 1 there is a path β ∈ U with β(0) = a and β(1) = b.…”

Section: A Path α : I → X Is Well-ended If For Every Open Neighborhoomentioning

confidence: 99%

Section: Proof If Ymentioning

confidence: 99%

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Semicoverings: a generalization of covering space theory

Brazas¹

2012

Homology, Homotopy and Applications

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“…We briefly mention that there are many natural approaches closely related to shape theory such as R.H. Fox's theory of overlays [25]. Similar approaches using pro-groups or other algebraic structures in place of the usual fundamental group are considered and compared in [1,28,29,37] and the references therein. Covering theories for categories other than the usual topological category, such as uniform spaces [3,10] and topological groups [2] have also appeared.…”

Section: Introductionmentioning

confidence: 99%

On maps with continuous path lifting

Brazas¹,

Mitra²

2020

Preprint

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We study a natural generalization of covering projections defined in terms of unique lifting properties. A map p : E → X has the "continuous path-covering property" if all paths in X lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally pathdisconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological π 1 : for any path-connected Hausdorff space X, maps E → X with the continuous path-covering property are classified up to weak equivalence by subgroups H ≤ π 1 (X, x 0 ) with totally path-disconnected coset space π 1 (X, x 0 )/H. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

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Fundamental groups and finite sheeted coverings

Paricio

Matijević

2010

Journal of Pure and Applied Algebra

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Communicated by E.M. Friedlander MSC: 55P57 55Q52 55Q70 55P55 55U40 55Q07 55R65 a b s t r a c tIt is well known that for a connected locally path-connected semi-locally 1-connected space X , there exists a bi-unique correspondence between the pointed d-fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σ d of degree d.The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the problem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different notions of 'fundamental group' have appeared in the mathematical literature, for instance, the Brown-Grossman-Quigley fundamental group, the Čech-Borsuk fundamental group, the Steenrod-Quigley fundamental group, the fundamental profinite group or the fundamental localic group.The main result of this paper determines different 'fundamental groups' that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X .

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Fundamental progroupoid and bundles with a structural category

Cited by 5 publications

References 5 publications

Semicoverings: a generalization of covering space theory

Semicoverings: a generalization of covering space theory

On maps with continuous path lifting

Fundamental groups and finite sheeted coverings

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